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Quadratic equations

Lesson 77 of 113

1. Ye topic hai kya?

Ek unknown x, power 2. Equation = 0 ke roots nikaalo — kitne real, equal ya nahi, formula se, word se quadratic banao.

Yahan lock

ax²+bx+c=0, split, square poora karo, x=(−b±√D)/2a, D=b²−4ac, nature, k for equal/real, consecutive / area word.

Yahan kya nahi

p(a), (x−a) factor as chapter — Polynomials (yahan sirf toot ke roots). Do variables do lines — Pair of linear. AP, pita–beta linear age — alag pages.

2. Standard form — pehle yahan lao

ax² + bx + c = 0, a ≠ 0.
a = quadratic coeff, b = linear, c = constant. Roots wo x jahan left 0.

Pehle sab left: x²=9 → x²−9=0. (x+1)(x−4)=2 → expand, =0. 1/x = x−1 → multiply x (x≠0): 1 = x²−x → x²−x−1=0.

LikhaQuadratic?Kyun
2x² − 7x + 3 = 0haana=2≠0
x² = 4xhaanx²−4x=0, c=0 allowed
3x − 5 = 0nahia=0, linear
x³ − x = 0nahidegree 3
(x−2)² = 0haanx²−4x+4=0, equal roots
x²=0 bhi quadratic (a=1, b=0, c=0), root 0 (repeated). “c nahi to quadratic nahi” galat.
2x=4 equation hai, polynomial quadratic nahi.

3. Factorise — ac aur b

Polynomials mein splitting seekha. Yahan equation: factors 0, har bracket se root. Alag example.

ax²+bx+c. Do numbers product ac, sum b. Split b, group, ( )( )=0.
Har factor 0 → do roots.
Example · split

2x² − 7x + 3 = 0. Roots?

Solution

  1. ac, b

    ac=6, b=−7. Numbers −6, −1. (−6)+(−1)=−7, product 6.

  2. Split

    2x² − 6x − x + 3 = 2x(x−3) −1(x−3) = (2x−1)(x−3)=0.

  3. Roots

    2x−1=0 → x=1/2. x−3=0 → x=3.

Answer1/2 aur 3

x²−8x=0 → x(x−8)=0, roots 0 aur 8. Constant 0 ho to x common.

Split na mile (D perfect square nahi / D<0) to formula. Integers ac-sum na hon to bhi formula chalega.

4. Completing the square — formula yahin se

x² + px ko poora square: aadha linear ka square jodo-ghatao. (x + p/2)² = x² + px + (p/2)².

Example · square

x² − 6x − 7 = 0.

Solution

  1. Constant right

    x² − 6x = 7.

  2. Aadha of 6

    3. 3²=9 dono taraf: x²−6x+9=16.

  3. Square

    (x−3)²=16. x−3=±4. x=7 ya x=−1.

Answer7, −1

Agar a≠1: pehle a se divide, phir same. 2x²−8x−10=0 → x²−4x−5=0 → (x−2)²=9 → x=5 ya −1.

5. Quadratic formula — D lock

Wahi square, general a,b,c pe:

x² + (b/a)x = −c/a. Jodo (b/2a)². Left (x + b/2a)². Right (b²−4ac)/(4a²).

D = b² − 4ac (discriminant)
x = (−b ± √D) / (2a)
√D tab real jab D ≥ 0. Denominator 2a, sirf a nahi.
Example · formula

x² − 4x − 5 = 0.

Solution

  1. D

    a=1, b=−4, c=−5. D=16 − 4(1)(−5)=16+20=36. √D=6.

  2. Roots

    [4 ± 6]/2. (10)/2=5, (−2)/2=−1.

Answer5, −1
b=−4 ho to −b = +4. (−b)² nahi, formula mein −b phir ±.
2a: a=2 ho to divide 4 se. √D ko 2a ke saath, a ke saath nahi.
D negative pe √D real nahi — Class-X “no real roots”. Complex i agla level.

6. Nature of roots — teen pictures

Parabola y=ax²+bx+c, roots = x-axis cuts. Pair-of-linear ki do seedhi lines nahi; Polynomials wale (x−1)(x−3) figure bhi nahi — yahan D se teen cases.

D > 0 · do alag

x²−2x−3=0. Do gold cuts.

D = 0 · equal

(x−2)²=0. Touch, do equal roots 2,2.

D < 0 · koi real nahi

x²−2x+5. Axis ke upar, cut zero.

D > 0 → do distinct real. Agar a,b,c rational aur D perfect square → roots rational; warna irrational (jaise √2).
D = 0 → real, equal (repeated). x = −b/(2a).
D < 0 → no real roots.

a>0 parabola upar khulti (U). a<0 neeche (∩) — D wahi rule, graph ulta.

k — equal / real / none

Pair-linear ka k slopes pe tha. Yahan ek equation, D mein k.

Example · k

x² − kx + 4 = 0. Equal roots ke liye k? Real roots?

Solution

  1. D

    k² − 16. Equal: D=0 → k²=16 → k=4 ya k=−4.

  2. Real

    D≥0 → k²≥16 → k≤−4 ya k≥4. None: −4<k<4 (D<0).

Answerequal k=±4; real |k|≥4
Equal ke liye D=0, D>0 nahi. “Real” mein equal shamil (D≥0) jab paper “at least one real” / “real roots” bole — do equal bhi real hain.

7. Sum / product — polynomials se uthake use

α, β roots of ax²+bx+c=0. Wahi: α+β=−b/a, αβ=c/a. Yahan kaam: check, naya expression, equation banao (monic).

α²+β²=(α+β)²−2αβ
Equal roots ⇒ α=β ⇒ (α+β)²=4αβ ⇒ b²=4ac — D=0 hi.

3x² − 2x − 1=0: sum=2/3, product=−1/3. (3x+1)(x−1)=3x²−2x−1. Roots −1/3, 1.

Equation banao zeros se: x²−(sum)x+(product)=0. Polynomials page pe 4,−1 tha. Yahan jaise 5 aur −2: x²−3x−10=0.

8. Word — quadratic banani padti hai

Unknown ek number / ek side. Product, area, consecutive — x(x+k)=m. Age (farq constant, linear ratio) Age page. Do keematein Pair of linear.

Sentence → =0 form → D/split → discard jo context tod-e (negative length, fraction people).
Example · consecutive

Do consecutive integers ka product 56. Chhota?

Solution

  1. n(n+1)=56

    n²+n−56=0. D=1+224=225=15².

  2. n

    (−1±15)/2. 14/2=7, −16/2=−8. Consecutive 7,8 ya −8,−7. “Chhota positive” 7.

Answer7 (saath 8)
Example · rectangle

Lambai, chaurai se 5 zyada. Area 36. Chaurai?

Solution

  1. b(b+5)=36

    b²+5b−36=0. D=25+144=169=13².

  2. b

    (−5±13)/2. 8/2=4, −18/2=−9. Length negative nahi → b=4, L=9. Check 4×9=36.

Answer4
Dono roots mathematically theek, context ek kaat-ta. Speed/time quadratic (tez to kam hours) TSD page ke uniform formula se alag — agar aaye to x(x+Δ)=distance setup, yahan area/consecutive kaafi LDC ke liye.

9. Sawal — basic se pro

Q1 · Basic

Kaun si quadratic equation hai?

  1. x³ − 4x = 0
  2. 2x − 7 = 0
  3. x² − 4x = 0
  4. xy = 6

Solution

  1. a≠0, degree 2, one var

    C: x(x−4)=0. A cubic. B linear. D two var (pair-linear type).

AnswerC
Q2 · Basic

x² − 7x + 10 = 0 ke roots?

  1. 2, 5
  2. −2, −5
  3. 1, 10
  4. 7, 10

Solution

  1. Split

    x²−2x−5x+10=(x−2)(x−5). B: x²+7x+10. C: sum 11≠7.

AnswerA · 2, 5
Q3 · Basic · D

2x² − 3x + 1 = 0 ka D?

  1. 1
  2. 9
  3. 5
  4. −1

Solution

  1. b²−4ac

    9 − 8=1. Roots (3±1)/4 → 1 aur 1/2.

AnswerA · 1
Q4 · Basic · nature

x² + x + 1 = 0 ke real roots?

  1. Do distinct
  2. Equal
  3. Koi nahi
  4. Ek

Solution

  1. D

    1−4=−3 < 0. No real. “Ek” nahi hota unless D=0 (wo bhi technically do equal).

AnswerC
Q5 · Medium · formula

x² − 4x − 5 = 0. Roots?

  1. 5, −1
  2. 4, 5
  3. −5, 1
  4. 5, 1

Solution

  1. D=36

    [4±6]/2 → 5, −1. C signs ulta. D: 5+1=6≠4 sum.

AnswerA
Q6 · Medium · k

x² − kx + 4 = 0 equal roots. k?

  1. sirf 4
  2. ±4
  3. ±2
  4. 16

Solution

  1. D=0

    k²−16=0, k=±4. A adha. C: D=4−16≠0.

AnswerB · ±4
Q7 · Medium · square

x² − 6x − 7 = 0 completing square se positive root?

  1. 3
  2. 7
  3. 6
  4. 1

Solution

  1. (x−3)²=16

    x=7 ya −1. Positive 7. 3 to square ka aadha hai, root nahi.

AnswerB · 7
Q8 · Medium

3x² − 2x − 1 = 0. Sum aur product of roots?

  1. sum 2/3, product −1/3
  2. sum −2/3, product −1/3
  3. sum 2, product −1
  4. sum 2/3, product 1/3

Solution

  1. −b/a, c/a

    2/3, −1/3. D: product sign. C: a se divide nahi. Roots 1, −1/3.

AnswerA
Q9 · Pro · word

Do consecutive integers, product 56. Bada wala (positive pair)?

  1. 7
  2. 8
  3. 6
  4. 9

Solution

  1. n(n+1)=56

    n=7, n+1=8. 8×7=56. D 9×6=54.

AnswerB · 8
Q10 · Pro

x² − 4x + k = 0 ke do distinct real roots. k?

  1. k = 4
  2. k > 4
  3. k < 4
  4. k ≥ 4

Solution

  1. D=16−4k

    Distinct: D>0 → 16−4k>0 → k<4. Equal k=4. None k>4.

AnswerC · k < 4

Practice

Pehle khud, phir neeche kholo.

12345 678910
P1

x² = 9 ko standard form?

  1. x² + 9 = 0
  2. x² − 9 = 0
  3. x − 3 = 0
  4. x² − 3 = 0

Solution

  1. Left

    x²−9=0. (x−3)(x+3)=0, roots ±3. C sirf ek root. D: x=±√3.

AnswerB
P2

x² − 8x = 0 ke roots?

  1. 8, 8
  2. 0, 8
  3. 0, −8
  4. 4, 4

Solution

  1. x(x−8)=0

    0 aur 8. D vertex −b/2a=4, roots nahi.

AnswerB
P3

(x−2)² = 0. Nature?

  1. Do distinct
  2. Equal real
  3. No real
  4. Linear

Solution

  1. x²−4x+4=0

    D=16−16=0. Root 2,2.

AnswerB
P4

2x² − 7x + 3 = 0. Bada root?

  1. 1/2
  2. 3
  3. 7
  4. 2

Solution

  1. (2x−1)(x−3)

    1/2 aur 3. Bada 3.

AnswerB · 3
P5

x² − kx + 4 = 0 real roots. k?

  1. |k| ≥ 4
  2. |k| > 4
  3. k = 4 only
  4. sab k

Solution

  1. k²≥16

    Equal |k|=4 bhi real. Distinct |k|>4. Real = A.

AnswerA
P6

Roots 5 aur −2. Monic equation?

  1. x² − 3x − 10 = 0
  2. x² + 3x − 10 = 0
  3. x² − 7x − 10 = 0
  4. x² − 3x + 10 = 0

Solution

  1. x²−(sum)x+prod

    Sum 3, product −10 → x²−3x−10. B sum −3. D product +10.

AnswerA
P7

Rectangle: lambai chaurai se 5 zyada, area 36. Lambai?

  1. 4
  2. 5
  3. 9
  4. 6

Solution

  1. b=4, L=9

    4×9=36. A chaurai hai. D 6×11=66.

AnswerC · 9
P8

a=1, b=−4, c=1. Roots formula se (simplify)?

  1. 2 ± √3
  2. 2 ± 1
  3. −2 ± √3
  4. 4 ± √3

Solution

  1. D=16−4=12=4·3

    [4±2√3]/2 = 2±√3. C: −b sign. B D=4 samajh liya.

AnswerA · 2±√3
P9

D>0, a,b,c rational, D=8. Roots?

  1. Rational
  2. Irrational real
  3. Not real
  4. Equal

Solution

  1. 8 perfect square nahi

    √8=2√2 irrational. Real distinct. Equal D=0. Rational D perfect square (0,1,4,9…).

AnswerB
P10

2x² − 8x − 10 = 0. Pehle a se divide, roots?

  1. 5, −1
  2. 4, 2
  3. 8, 10
  4. 5, 1

Solution

  1. ÷2

    x²−4x−5=0. (x−5)(x+1). Same roots original ke — divide se roots nahi badalte.

AnswerA · 5, −1

← Index · Pair of linear equations

Agle topic: Arithmetic progressions.